During my coursework in Philosophy, we devoted a lot of time to discussing Bayes’ theorem. Two fields find it particularly important, the Philosophy of Science and Epistemology, or the study of what knowledge is. It’s considered a pillar for rational thinking and increasing our understanding of the world, and it’s fundamental for evaluating claims given the evidence we have. Bayes’ theorem looks like this:
To put it simply, Bayes’ theorem describes the probability of an hypothesis or event based on relevant conditions or evidence. This equation might look complex, but it’s actually quite easy to understand after a little bit of translation. ‘P’ stands for ‘the probability that’, ‘|’ is a symbol that means something like ‘given that’, ‘A’ stands for a hypothesis, and ‘B’ stands for an event or evidence that might impact the likelihood of the hypothesis. When we understand it this way, the equation reads: the probability of a hypothesis given some evidence is equal to the probability of that evidence given the hypothesis, multiplied by the probability of hypothesis, and all of this is divided by the probability of that evidence.
An example can clear things up. Let’s say you check WebMD because you have a nasty cough. You see that having a nasty cough is a symptom of cancer, and that the likelihood of having this cough if you have cancer is very, very high. If you had cancer, this nasty cough is exactly what you would have expect to see, so it must be pretty probable that you have cancer, and like most people who visit WebMD, you walk away convinced that you’re dying. Bayes’ theorem helps us see why thinking this way is a mistake.
Let’s fill in the equation with some numbers we made up. Let’s assume the probability that you have the cough given that you have cancer is very high: 95%. But, you’re a young and healthy person, so at your age, only one in a hundred thousand people get this kind of cancer. And again, lets assume having a nasty cough is pretty common, it’s cold season after all, so one in a hundred people have a nasty cough. Filling it in, we get this:
So, if we do the math, we come up with your probability of having cancer given that you have a nasty cough: 0.00095, a pretty small chance.
The application of Bayes’ theorem in the field of medicine is extremely useful, especially when considering the accuracy of tests and the likelihood of false positives or false negatives, and there are countless other practical applications for it.
Bayes’ theorem is quite simple, but it’s application to the field of statistics, or Bayesian Statistics, is quite complex, and it’s an important part of how Google can filter search results for you, how your email can detect spam, and how Nate Silver could accurately predict the 2008 presidential election in the United States.
A PhD in Astronomy, Romke Bontekoe typically offers his course on Bayes in Action in Amsterdam, but on October 20th, he’ll be offering his training here at the European Data Innovation Hub. The training is geared towards managers and researchers who want to understand Bayesian Statistics and its application, but the course is open to anybody interested.
If you’d like to learn more about Bayes’ theorem, you can look at this video that I animated for Wireless Philosophy, and if you want to learn more about Bayesian Statistics and its application, register for the training on the di-academy’s website.